Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $p = \dfrac{10(2x - 5)}{3x} \div \dfrac{10(2x - 5)}{4x} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{10(2x - 5)}{3x} \times \dfrac{4x}{10(2x - 5)} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 10(2x - 5) \times 4x } { 3x \times 10(2x - 5) } $ $ p = \dfrac{40x(2x - 5)}{30x(2x - 5)} $ We can cancel the $2x - 5$ so long as $2x - 5 \neq 0$ Therefore $x \neq \dfrac{5}{2}$ $p = \dfrac{40x \cancel{(2x - 5})}{30x \cancel{(2x - 5)}} = \dfrac{40x}{30x} = \dfrac{4}{3} $